Compactness and Symmetric Well Orders
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866916150790586368 |
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| author | Dasgupta, Abhijit |
| author_facet | Dasgupta, Abhijit |
| contents | We introduce and investigate a topological version of Stäckel's 1907 characterization of finite sets, with the goal of obtaining an interesting notion that characterizes usual compactness (or a close variant of it). Define a $T_2$ topological space $(X, τ)$ to be Stäckel-compact if there is some linear ordering $\prec$ on $X$ such that every non-empty $τ$-closed set contains a $\prec$-least and a $\prec$-greatest element. We find that compact spaces are Stäckel-compact but not conversely, and Stäckel-compact spaces are countably compact. The equivalence of Stäckel-compactness with countable compactness remains open, but our main result is that this equivalence holds in scattered spaces of Cantor-Bendixson rank $< ω_2$ under ZFC. Under V=L, the equivalence holds in all scattered spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_13455 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Compactness and Symmetric Well Orders Dasgupta, Abhijit General Topology Logic 54D30 (Primary), 03E20, 03E65 (Secondary) We introduce and investigate a topological version of Stäckel's 1907 characterization of finite sets, with the goal of obtaining an interesting notion that characterizes usual compactness (or a close variant of it). Define a $T_2$ topological space $(X, τ)$ to be Stäckel-compact if there is some linear ordering $\prec$ on $X$ such that every non-empty $τ$-closed set contains a $\prec$-least and a $\prec$-greatest element. We find that compact spaces are Stäckel-compact but not conversely, and Stäckel-compact spaces are countably compact. The equivalence of Stäckel-compactness with countable compactness remains open, but our main result is that this equivalence holds in scattered spaces of Cantor-Bendixson rank $< ω_2$ under ZFC. Under V=L, the equivalence holds in all scattered spaces. |
| title | Compactness and Symmetric Well Orders |
| topic | General Topology Logic 54D30 (Primary), 03E20, 03E65 (Secondary) |
| url | https://arxiv.org/abs/2207.13455 |